$Show that the distance between two nonparallel lines is given by $\frac{|(p_2-p_1)\cdot (a_1\times a_2)|}{|| a_2\times a_1||}$$

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Show that the distance between two nonparallel lines is given by
$$\frac{|(p_2-p_1)\cdot (a_1\times a_2)|}{|| a_2\times a_1||}$$
where $p_1, p_2$ are any two points on the lines $l_1$ and $l_2$, and $a_1$ and $a_2$ are the directions of $l_1$ and $l_2$.

Calculus Multivariable Calculus Algebra

Karl Sch

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Let $P_1, P_2$ be the two parallel planes passing through $l_1, l_2$, respectively. Then the vector
\[n=a_1 \times a_2\]
is a normal vector for $P_1$ and $P_2$. Notice that The distance between the two lines is the same as the distance between the two parallel planes $P_1, P_2$. Since $p_1 \in P_1$ and $p_2 \in P_2$, the distance between the planes is the length of the orthogonal projection of the vector $(p_2-p_1)$ onto the normal vector $n$, i.e.
\[\text{Distance}=|| \frac{(p_2-p_1)\cdot (a_1\times a_2)}{||a_1\times a_2||^2} (a_1\times a_2)||\]
\[=| \frac{(p_2-p_1)\cdot (a_1\times a_2)}{||a_1\times a_2||^2}| || (a_1\times a_2)||\]
\[=\frac{|(p_2-p_1)\cdot (a_1\times a_2)|}{||a_1\times a_2||}.\]

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$Show that the distance between two nonparallel lines is given by $\frac{|(p_2-p_1)\cdot (a_1\times a_2)|}{|| a_2\times a_1||}$$

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